1. Section 5.5 - Independent Events Objectives: 1.Understand the definition of independent events. 2.Know how to use the Multiplication Rule for Independent Events. 3.Understand that real data rarely meet the mathematical definition of independence. 4.Develop insight into when it is reasonable to assume independence as part of the model.

2. Section 5.5 - Independent Events Example: Tap Water and Bottled Water In Jack and Jill’s tests with tap and bottled water, they found that how well a person did depended on whether that person regularly uses bottled water. Of the people who regularly drink bottled water, 24 / 30 = 80% correctly identified tap water. Of the people who drink tap water, only 36 / 70 = 51.4% correctly identified tap water. Display 5.48 Identified Tap Water YesNoTotal Drinks Bottled Water? Yes24630 No363470 Total6040100

3. Section 5.5 - Independent Events Example: Tap Water and Bottled Water On the other hand, men and women did equally well in identifying tap water Men: 21 / 35 = 60% correctly identified tap water. Women: 39 / 65 = 60% correctly identified tap water. Overall: 60 /100 = 60% correctly identified tap water. Display 5.49 Identified Tap Water YesNoTotal Gender Male211435 Female392665 Total6040100

4. Example: Tap Water and Bottled Water The events “drinks bottled water” and “correctly identifies tap water” are dependent events. The events “is a male” and “correctly identifies tap water” are independent events. Being male or female doesn’t change the probability that a person correctly identifies tap water. Display 5.49 Identified Tap Water YesNoTotal Gender Male60%40%100% Female60%40%100% Total60%40%100% Display 5.48 Identified Tap Water YesNoTotal Drinks Bottled Water? Yes80%20%100% No51.4%48.6%100% Total60%40%100% Section 5.5 - Independent Events

5. Definition of Independent Events

6. Section 5.5 - Independent Events Definition: If and Only If S is true if and only if T is true. S T If S is true, then T is true.S => T If T is true, then S is true.T => S Example: A man may vote if and only if he is has registered to vote. V R If a man may vote, then he has registered to vote. V => R If a man has registered to vote, then he may vote. R => V If a man may vote, then he is a U.S. citizen. V => C However, if a man is a U.S. citizen, it is not true that he may vote. (He must be registered.)

7. Section 5.5 - Independent Events The Multiplication Rule for Independent Events Two events A and B are independent if and only if More generally, events A 1, A 2, …, A n are independent if and only if

8. Section 5.5 - Independent Events The Multiplication Rule for Independent Events This is both a rule for computing probabilities: and a definition of independence

9. Section 5.5 - Independent Events The Multiplication Rule for Independent Events Example: Four Flips If you flip a fair coin four times, what is the probability of four heads?

10. Section 5.5 - Independent Events The Multiplication Rule for Independent Events Example: Four Flips If you flip a fair coin four times, what is the probability of four heads? HHHH HHHT HHTH HTHH THHH HHTT TTHH HTHT THTH HTTH THHT TTTT TTTH TTHT THTT HTTT P(HHHH) = 1/16

11. Section 5.5 - Independent Events The Multiplication Rule for Independent Events Example: Four Flips If you flip a fair coin four times, what is the probability of four heads? The outcomes of the first flips don’t change the probabilities on the remaining flips, so the flips are independent.

12. Section 5.5 - Independent Events The Multiplication Rule for Independent Events Example: Health Insurance About 30% of adults ages 19 to 29 don’t have health insurance. What is the chance that if you choose two adults from this age group at random, the first has insurance and the second doesn’t?

13. Section 5.5 - Independent Events The Multiplication Rule for Independent Events Example: Health Insurance About 30% of adults ages 19 to 29 don’t have health insurance. What is the chance that if you choose two adults from this age group at random, the first has insurance and the second doesn’t? A small sample is selected from a large population, so it is appropriate to model the outcomes as independent events.

14. Section 5.5 - Independent Events The Multiplication Rule for Independent Events Example: Health Insurance About 30% of adults ages 19 to 29 don’t have health insurance. What is the chance that if you choose two adults from this age group at random, the first has insurance and the second doesn’t? Display 5.50 Has Insurance? 2nd Young Adult NoYesTotal 1st Young Adult No0.090.210.30 Yes0.210.490.70 Total0.300.701.00

15. Section 5.5 - Independent Events The Multiplication Rule for Independent Events Example: Computing the Probability of “At Least One” Suppose you take a random sample of ten adults from this age group. What is the probability that at least one of them doesn’t have insurance?

16. Section 5.5 - Independent Events The Multiplication Rule for Independent Events Example: Computing the Probability of “At Least One” Suppose you take a random sample of ten adults from this age group. What is the probability that at least one of them doesn’t have insurance? The event “at least one doesn’t have insurance” is the complement of the event “all have insurance”

17. Section 5.5 - Independent Events Independence with Real Data Example: Independence and Baseball In a recent season, by July 1 the L.A. Dodgers had won 41 games and lost 37. The breakdown by day / night is shown below in Display 5.51. If one of these games is chosen at random, are the events “win” and “day game” independent? Won the Game? YesNoTotal Time of Game Day111021 Night302757 Total413778

18. Section 5.5 - Independent Events Independence with Real Data Example: Independence and Baseball Are the events “win” and “day game” independent? P(win) = 41 / 78 = 0.526 P(win | day game) = 11 / 21 = 0.524 Since the two probabilities are not exactly equal, the events are not independent.

19. Section 5.5 - Independent Events Independence with Real Data Example: Independence and Baseball Are the events “win” and “day game” independent? P(win) = 41 / 78 = 0.526 P(win | day game) = 11 / 21 = 0.524 Since the two probabilities are not exactly equal, the events are not independent. Given that the Dodgers played 21 day games, the two percentages couldn’t be any closer: (12 / 21 = 0.571). It is impossible for the two percentages to be the same! The correct conclusion is that if these results can be considered a random sample of games, there isn’t sufficient evidence to say that the events are not independent.